Chapter 8

In Problems 53-56, without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution. $$ \begin{array}{r} x_{1}+2 x_{2}-x_{3}=0 \\ 4 x_{1}-x_{2}+x_{3}=0 \\ 5 x_{1}+x_{2}-2 x_{3}=0 \end{array} $$

Without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution. $$ \begin{array}{r} x_{1}+x_{2}+x_{3}=0 \\ x_{1}-2 x_{2}+x_{3}=0 \\ -2 x_{1}+x_{2}-2 x_{3}=0 \end{array} $$

In Problems 53-56, without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution. $$ \begin{array}{r} x_{1}+x_{2}+x_{3}=0 \\ x_{1}-2 x_{2}+x_{3}=0 \\ -2 x_{1}+x_{3}-2 x_{3}=0 \end{array} $$

Without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution. $$ \begin{array}{r} x_{1}+x_{2}-x_{3}+x_{4}=0 \\ 5 x_{2}+2 x_{4}=0 \\ x_{1}+x_{3}-x_{4}=0 \\ 3 x_{1}+2 x_{2}-x_{3}+x_{4}=0 \end{array} $$

In Problems 53-56, without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution. $$ \begin{aligned} x_{1}+x_{2}-x_{3}+x_{4} &=0 \\ 5 x_{2}+2 x_{4} &=0 \\ x_{1}+x_{3}-x_{4} &=0 \\ 3 x_{1}+2 x_{2}-x_{3}+x_{4} &=0 \end{aligned} $$

Without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution. $$ \begin{array}{r} x_{1}+x_{2}-x_{3}+x_{4}=0 \\ x_{1}+x_{2}+x_{3}-x_{4}=0 \\ 2 x_{2}+x_{3}+x_{4}=0 \\ x_{2}-x_{3}-x_{4}=0 \end{array} $$

Encode the word ( \(\left.\begin{array}{llll}1 & 0 & 0 & 1\end{array}\right)\) using the Hamming \((7,4)\) code.

In Problems 53-56, without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution. $$ \begin{array}{r} x_{1}+x_{2}-x_{3}+x_{4}=0 \\ x_{1}+x_{2}+x_{3}-x_{4}=0 \\ 2 x_{2}+x_{3}+x_{4}=0 \\ x_{2}-x_{3}-x_{4}=0 \end{array} $$